How can I freeze part of object while applying simple deform(twist)?
I believe using vertex groups/weight painting would be the best way to accomplish this. In the screenshot below the selected vertices are added to a vertex group and plugged into the deform modifier:



Keep in mind that with weight painting, you can also have areas that are only partly influenced: (more flexibility)



Lastly, there are tons of resources on the web about vertex groups, here is one link to the blender manual.